For surjectivity, see [1]}. To prove this map is closed, we need to show that for a sequence of geodesics \(\lbrace \gamma _n\rbrace \)  in \(\mathcal {G}(\sigma ) \)  whose endpoints converge to some \(\lbrace x,y\rbrace \) , there exist a subsequence of \(\lbrace \gamma _n\rbrace \)  converging to a geodesic with endpoints \(\lbrace x,y\rbrace \) .
Since the endpoints of \(\gamma _n\)  converges, all \(\gamma _n\)  will pass through a common compact ball in \(\tilde{S}\) , see [2]}. Thus we can apply Arzela-Ascoli theorem and find a subsequence of \(\lbrace \gamma _n\rbrace \)  converging to a geodesic segment \(\delta \)  in this compact ball. If \(\delta \)  could be extended to a nonsingular geodesic \(\gamma \) , we are done. Otherwise, suppose \(\delta \)  meets a cone point. Take another compact ball containing this cone point and apply Arzela-Ascoli theorem again to find a subsequence of the previous subsequence, whose limit is an extension of \(\delta \) . Since there are only countably many cone points, we could use the diagonal method and find a subsequence converging to a geodesic \(\gamma \supset \delta \) . The continuity of \(\partial \)  implies that the endpoints of \(\gamma \)  are exactly \(\lbrace x,y\rbrace \) .
